Quadratic forms over polynomial extensions of rings of dimension one

Abstract

It was proved in [6] that if R is a Dedekind domain, in which 2 is invertible, then every quadratic space over R[X] which contains a hyperbolic space of rank two modulo X is extended from R. In this paper we give a classification of quadratic spaces over R[X] which contain a hyperbolic space of rank two modulo X, where R is any reduced commutative noetherian ring of dimension one in which 2 is invertible and which has finite normalisation R in its total quotient ring. Let c denote the conductor of R in R. We prove [Theorem I.31 that any quadratic space over R[X] which contains a hyperbolic space of rank two modulo X is extended from R upto a rank two quadratic space which becomes hyperbolic over R[X] and Ric[X]. In Section 2 we give a classification of such rank 2 spaces and deduce that these rank 2 spaces are hyperbolic if and only if their discriminant is (-1). In Section 3 [Theorem 3.31 we prove that if 4 is any quadratic space of rank 23 over R]XI , . . . ,X,,] which contains a hyperbolic space of rank two modulo (Xi, . . . ,X,,), then 4 is extended from R if its discriminant is extended from R. We deduce [Corollary 3.51 that W(R[X])=* W(R)@Disc R[X]/Disc R. We are grateful to R. Sridharan for the interest he has shown in this work. We are indebted to Professor H. Bass for critically going through the manuscript and suggesting improvements.